Holomorphic Disks and the Chord Conjecture

关于凸接触形式的强阿诺德弦猜想

On the strong Arnold chord conjecture for convex contact forms

本文证明了在严格凸嵌入到$\mathbb{R}^{2n}$中诱导出的接触形式下，强阿诺德弦猜想成立，并给出了当凸性条件不满足时的反例。

通过归一化容量证明强 Arnold 弦猜想

Strong Arnold chord conjecture via normalized capacities

本文利用归一化容量证明，动态凸环面区域附近的所有星形区域边界上的任意 Legendrian 纽结均具有端点不同的 Reeb 弦，并给出高维推广。

$\mathbb{R}^{4}$ 中一致凸域边界的强 Arnol'd 弦猜想

The strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in $\mathbb{R}^{4}$

利用 Oliver Edtmair 的椭球嵌入构造，证明了 $\mathbb{R}^{4}$ 中一致凸域边界的强 Arnol'd 弦猜想，并引入最终等变本质 Legendrian 概念，给出其弦长上界。

关于可序性与弦猜想

On orderability and the chord conjecture

该研究为具有接触不可序性和辛化刚性的新接触流形类证明了Arnol'd弦猜想，涵盖Brieskorn流形等，还给出最小Reeb弦长度的统一上界，方法关联了Mohnke构造与接触Hofer几何。

余法丛的弦猜想

The chord conjecture for conormal bundles

本文通过建立环绕Floer上同调与路径空间同调的同构，并结合扭曲Hurewicz定理，证明了闭流形余球丛中与余法丛同位素的Legendre子流形均满足Arnol'd弦猜想。

Punctured holomorphic curves and Lagrangian embeddings

We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant mod…

接触瞬子与Weinstein猜想及Arnold弦猜想的证明

Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture

本文基于接触Hamilton动力学与接触瞬子模空间理论，通过利用Legendrian接触瞬子上同调的基本类存在性、评估横截性以及接触同胚的Legendrian化几何构造，完整证明了Weinstein猜想和Arnold弦猜想。

Hamilton-Arnold Chord And Periodic Orbits

In this article, we first prove that every Hamilton flow has at least as many Hamilton- Arnold chords as a smooth function on the Legendre submanifold of zero first cohomology has critical points. Second, we prove that every Hamilton flow has at least as many close Hamilton orbits as a smooth round function on the close Hamilton manifold of zero first cohomology has critical circles which implies that the so called Arnold-Ginzburg or Seifert conjecture holds.

Viterbo's spectral bound conjecture for homogeneous spaces

We prove a conjecture of Viterbo about the spectral distance on the space of compact exact Lagrangian submanifolds of a cotangent bundle $T^*M$ in the case where $M$ is a compact homogeneous space: if such a Lagrangian submanifold is contained in the unit ball bundle of $T^*M$, its spectral distance to the zero section is uniformly bounded. This also holds for some immersed Lagrangian submanifolds if we take into account the length of the maximal Reeb chord.

Transversality via gauge transformations for pseudo holomorphic disks in projectivized vector bundles

We show that one can achieve transversality for lifts of holomorphic disks to a projectivized vector bundle by locally enlarging the structure group and considering the action of gauge transformations on the almost complex structure, which eases computation considerably. As an application we prove a special case of the Arnold-Givental conjecture.

Thomas-Yau conjecture and holomorphic curves

The main theme of this paper is the Thomas-Yau conjecture, primarily in the setting of exact, (quantitatively) almost calibrated, unobstructed Lagrangian branes inside Calabi-Yau Stein manifolds. In our interpretation, the conjecture is that Thomas-Yau semistability is equivalent to the existence of special Lagrangian representatives. We clarify how holomorphic curves enter this conjectural picture, through the construction of bordism currents between Lagrangians, and in the definition of the Solomon functional. Under some extra hypothesis, we shall prove Floer theoretic obstructions to the ex…

全纯 Floer 理论与 Donaldson-Thomas 不变量

Holomorphic Floer theory and Donaldson-Thomas invariants

本文提出全纯 Floer 理论的若干预期性质，并猜想其可将 Hitchin/Seiberg-Witten 可积系统与 Donaldson-Thomas 不变量及四维 N=2 理论的 BPS 谱联系起来。